This unit explores the connections between percentages, decimals, and fractions. It focuses on strategies for solving problems involving percentages and applies these strategies to real life contexts.
use double number lines, ratio tables, and converting to equivalent fractions to solve percentage problems
use a calculator to solve problems with percentages
Percentages are commonly used in real life. Many of these applications involve money. So here we look at percentages applied to sales’ discounts and to investment.
This unit gives practice in using percentages in a range of situations but it also shows the link between percentages, decimals and fractions. So before using this unit it would be good to make sure that the class is familiar with decimals and fractions.
Percentages relate numbers to 100. So that 50% actually means 50 out of 100. This means that 50% of a quantity (say, 40) can be calculated by multiplying the quantity by 50/100 (say, 40 x 50/100). In this situation, 50% is already being represented as a fraction, 50/100. So there is clearly a direct link between percentages and fractions. On the other hand, fractions can be written as decimals. After all, 50/100 is the same as 0.5. So 0.5 can be used in calculations with percentages as easily as the fraction 50/100 can.
Whether you use a fraction or a decimal to represent a percentage is entirely up to you. Your choice might be made on the grounds of what is most convenient. In a given problem would you rather multiply by 50/100 or 0.5 or 1/2? Perhaps the advantage of keeping the 100 there, at least initially, is to remind you where percentage concept comes from.
Today we explore the relationship between percentages and fractions.
- Ask the students what 50% means. Many of them will know that it is equivalent to one-half. Ask them what they think the % sign means. Point out, if necessary, that the sign is derived from the symbol /, meaning "out of", and the two zeros from one hundred.
- Get the students to fold a one-hundred (10 x 10) square in half and check that the area of one-half of the square is 50 out of 100. Represent this on a double number line and in a place value table:
- Tell the students that they are going to fold the same square into quarters. Ask them to predict what one-quarter and three-quarters will be as percentages. Have them fold the paper and then open it up to check the areas take up by these fractions. Add these to the number line and place value table:
- Write the following fractions on the board and ask the students to express each of them as percentages. Suggest that they can use any representation (100 square, double number line, place value table) that they think will help them.
1/5 3/5 3/2 (1 1/2) 9/4 (2 1/4) 1/10 1/8
After a suitable period of time discuss their solutions. Highlight strategies such as:
- use of equivalent fractions, like: 1/5 = 2/10 = 20/100 = 20%;
- use of double number lines as below:
As a result we can see, for instance, that 3/2 = 150%.
- Use of ratio tables:
So from this we can see that 1/8 = 12 1/2 %.
- To consolidate the students’ ability to convert from fractions to percentages and vice versa give them the following percentages to convert to fractions:
60% 90% 37.5% 175% 250% 387.5%
Students may use any of the previous strategies to solve these problems:
Today we solve percentage problems using a range of strategies which we encourage the students to share.
- Give the students this problem to solve:
Cecil has a job at the golf course collecting lost golf balls among the trees and water-traps. He is allowed to keep 25 % of the balls he finds. One day Cecil finds 48 golf balls. How many is he allowed to keep?
- Let the students solve the problem in small co-operative groups and gather the class together to share strategies. Key strategies are:
- use fractional equivalents to simplify the problem: 25% is equivalent to 1/4 so the problem becomes 1/4 x 48. Note that x means "of’’ in this case, as in "one-quarter of forty-eight";
- use the double number line or ratio table to model the problem:
- Set up the following puzzles using coloured cubes and opaque plastic jars. Tell the students that they need to work out how many cubes of each colour are in the jars. Stress the need to record their solutions:
Jar A: 30 cubes, 50% yellow, 20% green, 30% blue (15 yellow, 6 green, 9 blue)
Jar B: 36 cubes, 25% red, 75% blue (9 red, 27 blue)
Jar C 20 cubes, 20% black, 5% blue, 50% green, 25% yellow (4 black, 1 blue, 10 green, 5 yellow)
Jar D: 16 cubes, 12.5% white, 37.5% red, 50% orange (2 white, 6 red, 8 orange)
Jar E: 60 cubes, 10% yellow, 20% blue, 30% green, 40% red (6 yellow, 12 blue, 18 green, 24 red)
- Challenge the students to make up their own percentage jar problems for others to solve.
In this lesson we use the percentage key on the calculator to solve problems. We encourage the students to justify the reasonableness of the answers that they get.
- Ask the students if they know how to work out 25% of 28 using a calculator. If possible use a projected calculator so everyone can see and key in 28 x 25%. Challenge them to find 40% of 35 by first estimating and then performing the operation on the calculator (35 ´ 40%.)
- Pose this problem: "Is 50% of 25 more, less, or the same as, 25% of 50?" Tell the students that you want them to explain their answers. Discuss the models students use to solve the problem: Examples might include:
- use of equivalent fractions:
50% is 1/2, so 50% of 25 is one-half of 25;
25% is 1/4, so 25% of 50 is one-quarter of 50;
Since 50 is twice as much as 25, and one-quarter is half of one-half, the answers must be equal;
- use of operational order:
50 x 25% gives the same result as 25 x 50% as only the order of the factors is changed.
- use of matching double number lines:
- use of equivalent fractions:
- Pose problems like those below. Tell the students that they can use a calculator if they wish but suggest that thinking about each problem may be a more beneficial first step.
25% of what number is 12? (25% x ? = 12) (Answer: 25% of 48)
40% of what number is 14? (40% x ? = 14) (Answer: 40% of 35)
What percentage of 28 is 21? (? % x 28 = 21)(Answer: 75% of 28)
What percentage of 18 is 6? (? %x 18 = 6) (Answer: 33.33%of 18)
- Share the strategies that students use to solve these problems. Continue to describe them using common language like, "Change to equivalent fractions", "Use double number lines", and "Use tidy or unit fractions". An example of using tidy fractions is to solve, "40% of what number is 14", using the fact that 20% or one-fifth must be 7.
- Students might enjoy making up problems of these types for others to solve.
Today we apply our knowledge of percentages to shopping problems.
- Tell the students that they now have enough tools to attempt some problems that occur in real life and involve percentages. Point out the many shops identify the amount of discount in a sale in terms of percentage off. Consider the example of a 25% off sale:
Ask the students which arrow they think matches what you would pay in a 25% off sale. Look for explanations like, "You would pay 75%of the price since 100%- 25%= 75%".
- Move to specific examples from a 25% off sale. Use examples from sales brochures where possible, though you may wish to round the prices for easier calculation at this stage. Get the students to work out their solutions in any way they wish and share their strategies with the whole class. Here is the example of an article costing $34 at normal price:
- Provide the students with some advertising brochures from shops and tell them that they can choose five articles to purchase from the brochure they receive. Tell them that the store is having a 30% off sale and they must work out what the discounted price will be. Before beginning, ask them if they can think of a way to estimate the sale price easily. Some may suggest that 30% is close to one-third and that an easy estimate is to take one-third off the price. An article that normally cost $18 would now cost about one-third ($6) less, that is, $12.
- When the students have had sufficient time to work on these problems, put them in pairs to check each other’s calculations. You may wish to show them how the problems can be solved on a calculator. For example, the price of a $72 article could be worked out by keying in 72 x 30% (50.4). Be sure to ask for the interpretation of 50.4 in terms of the context ($50.40).
- To conclude the session, pose this problem. "Suppose you have $1000. You want to buy as many computer games as you can. Normally they cost $100 each. How many can you buy with no discount, 10% discount, 20% discount, 30% discount, etc? What do you notice about the number of games you can buy as the discount gets greater?" Suggest to the students that they may want to use a table or graph to record their findings.
Students may notice that the impact of discount is not linear. For example, a purchaser gets twice as many of an article at 70% discount as they do at 40% discount.
In today’s session we explore the use of percentages through banking problems.
- Explore investing money with the students. Tell them that they have $1 000 000 to invest and they must leave the money invested for ten years. There are four investment options for them:
SafeBank will pay you $100 000 every year for the whole ten years your money is with them
RegularBank will pay you 6% compound interest at the end of every year. That means the 6% you earned in the year before will also earn interest.
StepBank will pay you 1% compound interest at the end of the first year, 2% at the end of the second year, 3% at the end of the third year, and so on until they pay 10% at the end of the tenth year.
FiveYearBank will pay you 50% compound interest on your money but only every five years.
Which bank will give you the most money at the end of ten years?
- Students may wish to use a computer spreadsheet if this is available. Using formulae and the fill down function can save time and effort. If only calculators are available encourage them to organise their calculations in a table to help them see patterns:
- Ask the students to explain why they think some investment schemes offered better returns than others. The effect of compound interest can be demonstrated by getting them to graph the returns of each investment by year.
- Students might wish to investigate different investment deals that are available from local banks. These might include fixed and flexible interest rates.